vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...

, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following

differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

: :

\begin
  \nabla \times \mathbf &= \mathbf, \\
   \nabla \cdot \mathbf &= 0.
\end

From the vector calculus identity

\nabla^2 \mathbf \equiv \nabla (\nabla\cdot \mathbf) - \nabla\times (\nabla\times \mathbf)

it follows that :

\nabla^2 \mathbf = 0

that is, that the field v satisfies

Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...

. However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector field, which can be a point of confusion. For example, the vector field

= (xy, yz, zx)

satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a Laplacian vector field. A Laplacian vector field in the plane satisfies the

Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differenti ...

: it is holomorphic. Since the

curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was ...

of v is zero, it follows that (when the domain of definition is simply connected) v can be expressed as the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

of a

scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...

(see irrotational field) ''φ'' : :

\mathbf = \nabla \phi. \qquad \qquad (1)

Then, since the

divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...

of v is also zero, it follows from equation (1) that :

\nabla \cdot \nabla \phi = 0

which is equivalent to :

\nabla^2 \phi = 0.

Therefore, the potential of a Laplacian field satisfies

See also